'Hit the reset button' for Obamacare: Democrat

Lanny Davis, former Special Counsel to President Clinton, Independent Women's Forum, discuss health care in America. Heath says the payment system needs addressing.

In an effort to refocus public attention from Obamacare's website failures, the Obama administration is launching a two-pronged health care strategy this month to help promote the law's broader benefits.

President Barack Obama will focus on those benefits at a White House event Tuesday, trying to remind Americans that his health law is preventing insurance discrimination against those with pre-existing conditions and is allowing young people to stay on their parents' coverage until age 26. He'll also take aim at Republicans, arguing that the GOP is trying to strip away those benefits without presenting an alternative.

But it's not just Republicans voicing their concerns about Obamacare. Lanny Davis, a veteran Democratic strategist and former special counsel to President Clinton, told CNBC on Monday night that it might be time to start over.

Obama talks health care

CNBC's John Harwood reports President Obama is speaking to highlight the positives under the Affordable Care Act and is pushing enrollment on exchanges despite issues on the sites.

"I do concede that this is a mess. We may have tried too much, too fast and on a partisan basis," Davis told CNBC's "The Kudlow Report."

"We should hit the reset button and open ourselves up to some new ideas."

(Read more: Here's one thing right about Obamacare)

Behind the scenes, the administration is furiously trying to rectify an unresolved issue with enrollment data that could become a significant headache after the first of the year. Insurers say much of the enrollment data they're receiving are practically useless, meaning some consumers might not be able to get access to benefits on Jan. 1, the date their coverage is scheduled to take effect.

The website troubles resulted in significantly lower enrollment than what administration officials had hoped for and it's questionable whether the program will reach the 7 million sign-ups predicted by the Congressional Budget Office. The sign-up period runs through March 31.

The White House's new strategy comes as Obamacare's failures have put Democratic lawmakers facing re-election on edge, contributed to a drop in Obama's overall job approval rating, and emboldened Republicans.

Some in the GOP have called for a market-based fix for America's health care system, arguing that competition and market forces would be a more efficient use of health care dollars. When providers have to compete on price, Republicans say, prices become more transparent and lower. Their proposed changes include eliminating disadvantages in the tax code, rolling back coverage mandates and focusing government intervention only on the sickest and poorest.

Davis, author of the book "Crisis Tales: Five Rules for Coping with Crises in Business, Politics, and Life" agrees.

(Read more: Obama offers aid for lost coverage)

"I've always felt that going to a physician and taking tests that someone else is paying for, and you don't really care what's on the bill because it's a third-party payer rather than requiring people to have deductibles, where they at least have to know what they're getting and decide what they have to pay. That's the market operating. And it's a fundamental flaw with Obamacare."

"The reason that Bill Clinton was so successful is that he was a progressive Democrat on social and cultural issues," Davis told Larry Kudlow. "But he also believes that the free-market and business can be helpful to those progressive goals."

—By CNBC's Ben Thompson. Follow him on Twitter @BenThompson00. The Associated Press contributed to this report.